Slides by John Loucks St. Edward’s University

1 Slide

© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Slides by

JohnLoucks

St. Edward’sUniversity

Chapter 5 Advanced Linear Programming

Applications Data Envelopment Analysis

• Comparing performance of one branch to the whole

Revenue Management• Maximize revenue from short-term

demand of a fixed-inventory Portfolio Models and Asset Allocation

• Maximize return from a mix of investments

Game Theory• Compete with another player for a fixed

sum (ie market share)

3 Slide


Data Envelopment Analysis Data envelopment analysis (DEA)

• used to determine the relative operating efficiency of units with the same goals and objectives.

• Branch of a bank, franchise of a restaurant, etc

DEA creates a hypothetical composite• weighted average (W1, W2,…) of existing

units.• Optimal weights determined by the analysis

Goal is to determine E, the efficiency index for one unit• If E < 1, unit is less efficient than the

composite and be deemed relatively inefficient.

• If E = 1, unit is believed to be efficient compared to the rest.

4 Slide


Data Envelopment Analysis The DEA Model

MIN Es.t. OUTPUTS

INPUTSSum of weights = 1E, weights > 0

5 Slide


Data Envelopment Analysis Hospital Administrators at four hospitals want

to examine performance General, University, County, State Inputs:

• Number of FTE nonphysician personnel• Amount spent on supplies• Number of bed-days available

Outputs• Patient days of Medicare Service• Patient days of non-Medicare Service• Number of nurses trained• Number of interns trained

6 Slide


Data Envelopment Analysis

Input

Output

7 Slide


Data Envelopment Analysis Output

8 Slide


Data Envelopment Analysis Define the Decision Variables

E = Fraction of County's input resources required, compared to the composite hospitalwg = Weight applied to General hospitalwu = Weight applied to University Hospitalwc = Weight applied to County Hospitalws = Weight applied to State Hospital

9 Slide


Data Envelopment Analysis Define the Objective FunctionSince our objective is to detect inefficiencies,

Minimize the fraction of County’s input resources required by the composite high school:MIN E

10 Slide


Data Envelopment Analysis Define the Constraints

• Sum of the Weights is 1:• wg + wu + wc + ws = 1

Output Constraints• General form for each output:

• output for composite >= output for county• Output for composite =

• (Output for general * weight for general) +(output for university * weight for university) + (output for county * weight for county) +(output for state * weight for state)

11 Slide



Output Constraint for Medicare Patient Days• 48.14wg + 34.62wu + 36.72wc + 33.16ws

>= 36.72 Output Constraint for non-Medicare Patient

Days• 43.10wg + 27.11wu + 45.98wc + 56.46 ws

>= 45.98 Output constraint for nurses

• 253wg + 148wu + 175wc + 160ws >= 175 Output constraint for Interns

• 41wg + 27wu + 23wc + 84ws >= 23

12 Slide


Data Envelopment Analysis Input Constraints:

• General Form• Input for composite <= input for county * E

• Input for composite• (input for general * weight for general) +

(input for university * weight for university) +(input for county * weight for county) +(input for state * weight for state)

Nonnegativity of variables: E, w1, w2, w3 > 0

13 Slide



Input Constraint for FTE non-Physicians• 285.2wg + 162.3wu + 275.07wc + 210.4ws

<=275.7E Input constraint for supply expense

• 123.8wg + 128.7wu + 348.5wc + 154.1ws <= 348.5E

Input constraint for bed-days• 106.72wg + 64.21wu + 104.1wc + 104.04ws

<= 104.1E

14 Slide


Data Envelopment Analysis Computer Solution

OBJECTIVE FUNCTION VALUE = 0.765 VARIABLE VALUE REDUCED COSTS

E 0.765 0.000 W1 0.000

0.235 W2 0.500

0.000 W3 0.500

0.000

15 Slide


Data Envelopment Analysis Computer Solution (continued)

CONSTRAINT SLACK/SURPLUS DUAL VALUES

1 0.000 -0.235

2 65.000 0.000

3 0.000 -0.001

4 170.000 0.000

5 4.294 0.000

6 0.044 0.000

7 0.000 0.001

16 Slide


Data Envelopment Analysis Conclusion

The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)

17 Slide


Revenue Management Another LP application is revenue

management. Revenue management involves managing the

short-term demand for a fixed perishable inventory in order to maximize revenue potential.

The methodology was first used to determine how many airline seats to sell at an early-reservation discount fare and many to sell at a full fare.

Application areas now include hotels, apartment rentals, car rentals, cruise lines, and golf courses.

18 Slide


Revenue Management

LeapFrog Airways provides passenger service forIndianapolis, Baltimore, Memphis, Austin, and Tampa.LeapFrog has two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morningthe Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies toTampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity.

19 Slide


LeapFrog uses two fare classes: a discount fare Dclass and a full fare F class. Leapfrog’s products, eachreferred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares andforecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF.

Revenue Management

20 Slide


ODIF123456789

10111213141516

OriginIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sIndianapoli

sBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreBaltimoreMemphisMemphisMemphisMemphis

DestinationMemphis

AustinTampa

MemphisAustinTampa

MemphisAustinTampa

MemphisAustinTampaAustinTampa AustinTampa

FareClass

DDDFFFDDDFFFDDFF

ODIFCodeIMDIADITDIMFIAFITF

BMDBADBTDBMFBAFBTFMADMTDMAFMTF

Fare175275285395425475185315290385525490190180310295

Demand

44254015108

26504212169

58481411

Revenue Management

21 Slide


Revenue Management Define the Decision VariablesThere are 16 variables, one for each ODIF:IMD = number of seats allocated to Indianapolis-Memphis-

Discount classIAD = number of seats allocated to Indianapolis-Austin- Discount classITD = number of seats allocated to Indianapolis-Tampa- Discount classIMF = number of seats allocated to Indianapolis-Memphis- Full Fare classIAF = number of seats allocated to Indianapolis-Austin-Full Fare class

22 Slide


Revenue Management Define the Decision Variables (continued)ITF = number of seats allocated to Indianapolis-Tampa- Full Fare classBMD = number of seats allocated to Baltimore-Memphis- Discount classBAD = number of seats allocated to Baltimore-Austin- Discount classBTD = number of seats allocated to Baltimore-Tampa- Discount classBMF = number of seats allocated to Baltimore-Memphis- Full Fare classBAF = number of seats allocated to Baltimore-Austin- Full Fare class

23 Slide


Revenue Management Define the Decision Variables (continued)BTF = number of seats allocated to Baltimore-Tampa- Full Fare classMAD = number of seats allocated to Memphis-Austin- Discount classMTD = number of seats allocated to Memphis-Tampa- Discount classMAF = number of seats allocated to Memphis-Austin- Full Fare classMTF = number of seats allocated to Memphis-Tampa- Full Fare class

24 Slide


Revenue Management Define the Objective Function

Maximize total revenue: Max (fare per seat for each ODIF) x (number of seats allocated to the

ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF +

490BTF + 190MAD + 180MTD + 310MAF +

295MTF

25 Slide


Revenue Management Define the Constraints

There are 4 capacity constraints, one for each flight leg:

Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120

Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF

< 120 Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF <

120 Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF <

120

26 Slide


Revenue Management Define the Constraints (continued)

There are 16 demand constraints, one for each ODIF:

(5) IMD < 44 (11) BMD < 26 (17) MAD < 5

(6) IAD < 25 (12) BAD < 50 (18) MTD < 48

(7) ITD < 40 (13) BTD < 42 (19) MAF < 14

(8) IMF < 15 (14) BMF < 12 (20) MTF < 11

(9) IAF < 10 (15) BAF < 16 (10) ITF < 8 (16) BTF < 9

27 Slide


Revenue Management Computer Solution Objective Function Value = 94735.000

Variable Value Reduced Cost

IMD 44.000 0.000 IAD 3.000 0.000 ITD 40.000

0.000 IMF 15.000

0.000 IAF 10.000

0.000 ITF 8.000

0.000 BMD 26.000

0.000 BAD 50.000

0.000

28 Slide


Revenue Management Computer Solution (continued)

Variable Value Reduced Cost

BTD 7.000 0.000

BMF 12.000 0.000

BAF 16.000 0.000

BTF 9.000 0.000

MAD 27.000 0.000

MTD 45.000 0.000

MAF 14.000 0.000

MTF 11.000 0.000

29 Slide


Portfolio Models and Asset Management Asset allocation involves determining how to

allocate investment funds across a variety of asset classes such as stocks, bonds, mutual funds, real estate.

Portfolio models are used to determine percentage of funds that should be made in each asset class.

The goal is to create a portfolio that provides the best balance between risk and return.

30 Slide


John Sweeney is an investment advisor who isattempting to construct an "optimal portfolio" for aclient who has $400,000 cash to invest. There are tendifferent investments, falling into four broad categories that John and his client have identified as potential candidate for this portfolio. The investments and their important characteristics are listed in the table on the next slide. Note that Unidyde Corp. under Equities and Unidyde Corp. under Debt are two separate investments, whereas First General REIT is a single investment that is considered both an equities and a real estate investment.

Portfolio Model

31 Slide


Portfolio Model Exp. Annual

After Tax Liquidity RiskCategory Investment Return Factor FactorEquities Unidyde Corp. 15.0% 100 60(Stocks) CC’s Restaurants 17.0% 100 70 First General REIT 17.5% 100 75Debt Metropolis Electric 11.8% 95 20(Bonds) Unidyde Corp. 12.2% 92 30 Lewisville Transit 12.0% 79 22Real Estate Realty Partners 22.0% 0 50 First General REIT ( --- See above --- )Money T-Bill Account 9.6% 80 0 Money Mkt. Fund 10.5% 100 10 Saver's Certificate 12.6% 0 0

32 Slide


Portfolio Model

Formulate a linear programming problem toaccomplish John's objective as an investment advisor which is to construct a portfolio that maximizes hisclient's total expected after-tax return over the next year, subject to the limitations placed upon him by the client for the portfolio. (Limitations listed on next two slides.)

33 Slide


Portfolio Model

Portfolio Limitations 1. The weighted average liquidity factor for the portfolio must to be at least 65. 2. The weighted average risk factor for the portfolio must be no greater than 55. 3. No more than $60,000 is to be invested in Unidyde stocks or bonds. 4. No more than 40% of the investment can be in any one category except the money category. 5. No more than 20% of the total investment can be in any one investment except the money market fund.

continued

34 Slide


Portfolio Model

Portfolio Limitations (continued) 6. At least $1,000 must be invested in the Money Market fund. 7. The maximum investment in Saver's Certificates is $15,000. 8. The minimum investment desired for debt is $90,000. 9. At least $10,000 must be placed in a T-Bill account.

35 Slide


Portfolio Model Define the Decision Variables

X1 = $ amount invested in Unidyde Corp. (Equities)X2 = $ amount invested in CC’s RestaurantsX3 = $ amount invested in First General REITX4 = $ amount invested in Metropolis ElectricX5 = $ amount invested in Unidyde Corp. (Debt) X6 = $ amount invested in Lewisville TransitX7 = $ amount invested in Realty PartnersX8 = $ amount invested in T-Bill Account X9 = $ amount invested in Money Mkt. Fund

X10 = $ amount invested in Saver's Certificate

36 Slide


Portfolio Model Define the Objective Function

Maximize the total expected after-tax return over the next year:Max .15X1 + .17X2 + .175X3 + .118X4 + .122X5

+ .12X6 + .22X7 + .096X8 + .105X9 + .126X10

37 Slide


Portfolio Model

Total funds invested must not exceed $400,000:(1) X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 +

X10 = 400,000Weighted average liquidity factor must to be at

least 65:(2) 100X1 + 100X2 + 100X3 + 95X4 + 92X5 + 79X6 +

80X8 + 100X9 > 65(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9

+ X10)Weighted average risk factor must be no greater

than 55:(3) 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7

+ 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9

+ X10)No more than $60,000 to be invested in Unidyde

Corp:(4) X1 + X5 < 60,000

Define the Constraints

38 Slide


Portfolio Model Define the Constraints

(continued)No more than 40% of the $400,000 investment can be

in any one category except the money category:(5) X1 + X2 + X3 < 160,000(6) X4 + X5 + X6 < 160,000(7) X3 + X7 < 160,000No more than 20% of the $400,000 investment

can bein any one investment except the money market

fund:(8) X2 < 80,000 (12) X7 < 80,000(9) X3 < 80,000 (13) X8 < 80,000(10) X4 < 80,000 (14) X10 < 80,000(11) X6 < 80,000

39 Slide


Portfolio Model Define the Constraints

(continued)At least $1,000 must be invested in the Money Market fund:

(15) X9 > 1,000

The maximum investment in Saver's Certificates is $15,000:

(16) X10 < 15,000 The minimum investment the Debt category is

$90,000:(17) X4 + X5 + X6 > 90,000 At least $10,000 must be placed in a T-Bill account:(18) X8 > 10,000Non-negativity of variables: Xj > 0 j = 1, . . . , 10

40 Slide


Portfolio Model Solution Summary

Total Expected After-Tax Return = $64,355X1 = $0 invested in Unidyde Corp. (Equities)X2 = $80,000 invested in CC’s RestaurantsX3 = $80,000 invested in First General REITX4 = $0 invested in Metropolis ElectricX5 = $60,000 invested in Unidyde Corp. (Debt) X6 = $74,000 invested in Lewisville TransitX7 = $80,000 invested in Realty PartnersX8 = $10,000 invested in T-Bill Account X9 = $1,000 invested in Money Mkt. Fund

X10 = $15,000 invested in Saver's Certificate

41 Slide


Introduction to Game Theory In decision analysis, a single decision maker

seeks to select an optimal alternative. In game theory, there are two or more decision

makers, called players, who compete as adversaries against each other.

It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view.

Each player selects a strategy independently without knowing in advance the strategy of the other player(s).

continue

42 Slide


Introduction to Game Theory The combination of the competing strategies

provides the value of the game to the players. Examples of competing players are teams,

armies, companies, political candidates, and contract bidders.

43 Slide


Two-person means there are two competing players in the game.

Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player.

The gain and loss balance out so that there is a zero-sum for the game.

What one player wins, the other player loses.

Two-Person Zero-Sum Game

44 Slide


Competing for Vehicle SalesSuppose that there are only two vehicle

dealer-ships in a small city. Each dealership is consideringthree strategies that are designed to take sales of new vehicles from the other dealership over afour-month period. The strategies, assumed to be the same for both dealerships, are on the next slide.

Two-Person Zero-Sum Game Example

45 Slide


Strategy Choices Strategy 1: Offer a cash rebate

on a new vehicle. Strategy 2: Offer free optional

equipment on a new vehicle.

Strategy 3: Offer a 0% loan on a new vehicle.


46 Slide


2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

Payoff Table: Number of Vehicle Sales Gained Per Week by

Dealership A (or Lost Per Week by

Dealership B)

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A


47 Slide


Step 1: Identify the minimum payoff for each row (for Player A).

Step 2: For Player A, select the strategy that provides

the maximum of the row minimums (called

the maximin).


48 Slide


Identifying Maximin and Best Strategy

RowMinimum

1-3-2

2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Best Strategy

For Player AMaximinPayoff


49 Slide


Step 3: Identify the maximum payoff for each column

(for Player B). Step 4: For Player B, select the strategy that

provides the minimum of the column

maximums (called the minimax).


50 Slide


Identifying Minimax and Best Strategy

2 2 1

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Column Maximum 3 3 1

Best Strategy

For Player B

MinimaxPayoff


51 Slide


Pure Strategy

Whenever an optimal pure strategy exists: the maximum of the row minimums equals

the minimum of the column maximums (Player A’s maximin equals Player B’s minimax)

the game is said to have a saddle point (the intersection of the optimal strategies)

the value of the saddle point is the value of the game

neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategy

52 Slide


RowMinimum

1-3-2

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

-3 3 -1 3 -2 0

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A

Column Maximum 3 3 1

Pure Strategy Example

Saddle Point and Value of the Game

2 2 1

SaddlePoint

Value of thegame is 1

53 Slide


Pure Strategy Example

Pure Strategy Summary Player A should choose Strategy a1 (offer a

cash rebate). Player A can expect a gain of at least 1

vehicle sale per week. Player B should choose Strategy b3 (offer a

0% loan). Player B can expect a loss of no more than

1 vehicle sale per week.

54 Slide


Mixed Strategy

If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game.

In this case, a mixed strategy is best. With a mixed strategy, each player employs

more than one strategy. Each player should use one strategy some of

the time and other strategies the rest of the time.

The optimal solution is the relative frequencies with which each player should use his possible strategies.

55 Slide


Competing for Vehicle Sales Let us continue with the two-dealership

gamepresented earlier, but with a change to one payoff.If both Dealership A and Dealership B choose to offer a 0% loan, the payoff to Dealership A is nowan increase of 3 vehicle Sales per week. (The revised payoff table appears on the next slide.)

Two-Person Zero-Sum Game Example #2

56 Slide


Column Maximum

2 2 11

CashRebate

b1

0%Loan

b3

FreeOptions

b2

Dealership B

Payoff Table: Number of Vehicle Sales Gained Per Week by

Dealership A (or Lost Per Week by

Dealership B)

-3 3 -1-3 3 -2 3

-2

Cash Rebate a1

Free Options a2

0% Loan a3

Dealership A


3 3 3

Row min

57 Slide


Let us first consider the game from the point of view of Dealership A.

Dealership A will select one of its three strategies based on the following probabilities:PA1 = the probability that Dealership A selects strategy a1

PA2 = the probability that Dealership A selects strategy a2 PA3 = the probability that Dealership A selects strategy a3


58 Slide


Two-Person Zero-Sum Game Example #2 The expected gain under a specific competitor’s

strategy is a weighted average of: The gain under each of your strategies, times… The probability that you will choose said

strategy… Repeated For each of your competitor’s strategy

Dealership B Strategy Expected Gain for Dealership A

b1 EG(b1) = 2PA1 – 3PA2 + 3PA3

b2 EG(b2) = 2PA1 + 3PA2 – 2PA3 b3 EG(b3) = 1PA1 – 1PA2 + 3PA3

59 Slide


Two-Person Zero-Sum Game Example #2 Define GAINA to be the optimal expected gain in

vehicle sales for Dealership A, which we want to maximize.

Thus, the individual expected gains, EG(b1), EG(b2) and EG(b3) must all be greater than or equal to GAINA.

For example,2PA1 – 3PA2 + 3PA3 > GAINA

Also, the sum of Dealership A’s mixed strategy probabilities must equal 1.

This results in the LP formulation on the next slide …..

60 Slide


Dealership A’s Linear Programming FormulationMax GAINA

s.t.2PA1 – 3PA2 + 3PA3 – GAINA > 0

(Strategy b1)2PA1 + 3PA2 – 2PA3 – GAINA > 0 (Strategy

b2) 1PA1 – 1PA2 + 0PA3 – GAINA > 0 (Strategy b3)

PA1 + PA2 + PA3 = 1 (Prob’s sum to 1)

PA1, PA2, PA3, GAINA > 0 (Non-negativity)


61 Slide


Computer Solution: Dealership AOBJECTIVE FUNCTION VALUE = 1.333

VARIABLE VALUE REDUCED COSTS PA1 (cash) 0.833

0.000 PA2 (opts) 0.000 1.000 PA3 (0%) 0.167

0.000 GAINA 1.333 0.000


62 Slide


Computer Solution: Dealership ACONSTRAINT SLACK/SURPLUS DUAL VALUES

1 0.833 0.000

2 0.000 -0.333

3 0.000 -0.667

4 0.000 1.333


63 Slide


Dealership A’s Optimal Mixed Strategy• Offer a cash rebate (a1) with a probability of

0.833• Do not offer free optional equipment (a2)• Offer a 0% loan (a3) with a probability of 0.167

The expected value of this mixed strategy is a gain of

1.333 vehicle sales per week for Dealership A.


64 Slide


Let us now consider the game from the point of view of Dealership B.

Dealership B will select one of its three strategies based on the following probabilities:PB1 = the probability that Dealership B selects strategy b1

PB2 = the probability that Dealership B selects strategy b2 PB3 = the probability that Dealership B selects strategy b3


65 Slide


Because the numbers are reversed, we calculate the Expected Loss, instead of Expected GainDealership A Strategy Expected Loss for Dealership B

a1 EL(a1) = 2PB1 + 2PB2 + 1PB3

a2 EL(a2) = -3PB1 + 3PB2 – 1PB3 a3 EL(a3) = 3PB1 – 2PB2 + 3PB3


66 Slide


Two-Person Zero-Sum Game Example #2 Define LOSSB to be the optimal expected loss in

vehicle sales for Dealership B, which we want to MINIMIZE.

Thus, the individual expected losses, EL(a1), EL(a2) and EL(a3) must all be less than or equal to LOSSB.

For example,2PA1 + 2PA2 + 1PA3 < LOSSB

Also, the sum of Dealership B’s mixed strategy probabilities must equal 1.

This results in the LP formulation on the next slide …..

67 Slide


Dealership B’s Linear Programming FormulationMin LOSSB

s.t. 2PB1 + 2PB2 + 1PB3 – LOSSB < 0

(Strategy a1)-3PB1 + 3PB2 – 1PB3 – LOSSB < 0

(Strategy a2) 3PB1 – 2PB2 + 3PB3 – LOSSB < 0 (Strategy a3)

PB1 + PB2 + PB3 = 1 (Prob’s sum to 1)

PB1, PB2, PB3, LOSSB > 0 (Non-negativity)


68 Slide


Computer Solution: Dealership BOBJECTIVE FUNCTION VALUE = 1.333

VARIABLE VALUE REDUCED COSTS PB1 0.000 0.833

PB2 0.333 0.000 PB3 0.667 0.000 LOSSB 1.333 0.000


69 Slide


Computer Solution: Dealership BCONSTRAINT SLACK/SURPLUS DUAL VALUES

1 0.000 0.833

2 1.000 0.000

3 0.000 0.167

4 0.000 -1.333


70 Slide


Dealership B’s Optimal Mixed Strategy• Do not offer a cash rebate (b1)• Offer free optional equipment (b2) with a

probability of 0.333• Offer a 0% loan (b3) with a probability of 0.667

The expected payoff of this mixed strategy is a loss of1.333 vehicle sales per week for Dealership B.

Note that expected loss for Dealership B is the same asthe expected gain for Dealership A. (There is a

zero-sum for the expected payoffs.)


71 Slide


End of Chapter 5

Slides by John Loucks St. Edward’s University

Documents

cengage learning

accessible website

university output

county output

general output

output constraintsgeneral

wg wu wc ws

county hospitalws